A class has three teachers, Mr. $$P$$, Ms. $$Q$$ and $$Mrs. R$$ and six students $$A, B, C, D, E, F$$. Number of ways in which they can be seated in a line of $$9$$ chairs, if between any two teachers there are exactly two students, is $$k!(18)$$, then the value of $$k$$ is.
Let $$T$$ and $$S$$ denotes teacher and student respectively.
Then we have following possible patterns according to question
(i) $$T\ S\ S\ T\ S\ S\ T\ S\ S$$
(ii) $$S\ T\ S\ S\ T\ S\ S\ T\ S$$
(iii) $$S\ S\ T\ S\ S\ T\ S\ S\ T$$
Hence total number of arrangements are $$3 \cdot (3!) 6! = 18\times 6! \Rightarrow k = 6$$.